LMFDB - Embedded newform 1521.2.b.g.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1351,1521)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1521.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 25 $ x^4 - 5*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$1.93649 + 1.11803i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.g.1351.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{2} -3.00000 q^{4} -2.23607i q^{5} -3.46410i q^{7} +2.23607i q^{8} -5.00000 q^{10} +4.47214i q^{11} -7.74597 q^{14} -1.00000 q^{16} -3.87298 q^{17} -3.46410i q^{19} +6.70820i q^{20} +10.0000 q^{22} -7.74597 q^{23} +10.3923i q^{28} +3.87298 q^{29} +6.70820i q^{32} +8.66025i q^{34} -7.74597 q^{35} -1.73205i q^{37} -7.74597 q^{38} +5.00000 q^{40} -2.23607i q^{41} +2.00000 q^{43} -13.4164i q^{44} +17.3205i q^{46} +4.47214i q^{47} -5.00000 q^{49} -11.6190 q^{53} +10.0000 q^{55} +7.74597 q^{56} -8.66025i q^{58} -8.94427i q^{59} -7.00000 q^{61} +13.0000 q^{64} +3.46410i q^{67} +11.6190 q^{68} +17.3205i q^{70} +4.47214i q^{71} -15.5885i q^{73} -3.87298 q^{74} +10.3923i q^{76} +15.4919 q^{77} +8.00000 q^{79} +2.23607i q^{80} -5.00000 q^{82} +4.47214i q^{83} +8.66025i q^{85} -4.47214i q^{86} -10.0000 q^{88} +4.47214i q^{89} +23.2379 q^{92} +10.0000 q^{94} -7.74597 q^{95} +6.92820i q^{97} +11.1803i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{4} - 20 q^{10} - 4 q^{16} + 40 q^{22} + 20 q^{40} + 8 q^{43} - 20 q^{49} + 40 q^{55} - 28 q^{61} + 52 q^{64} + 32 q^{79} - 20 q^{82} - 40 q^{88} + 40 q^{94}+O(q^{100}) $ 4 * q - 12 * q^4 - 20 * q^10 - 4 * q^16 + 40 * q^22 + 20 * q^40 + 8 * q^43 - 20 * q^49 + 40 * q^55 - 28 * q^61 + 52 * q^64 + 32 * q^79 - 20 * q^82 - 40 * q^88 + 40 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.23607i	− 1.58114i	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	− 2.23607i	− 1.58114i	0.612372	−	0.790569i	$-0.290215\pi$
$3$	0	0
$3$	0	0
$4$	−3.00000	−1.50000
$5$	− 2.23607i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$5$	− 2.23607i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$6$	0	0
$7$	− 3.46410i	− 1.30931i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	− 3.46410i	− 1.30931i	0.755929	−	0.654654i	$-0.227186\pi$
$8$	2.23607i	0.790569i
$9$	0	0
$10$	−5.00000	−1.58114
$11$	4.47214i	1.34840i	0.738549	+	0.674200i	$0.235511\pi$
$11$	4.47214i	1.34840i	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−7.74597	−2.07020
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−3.87298	−0.939336	−0.469668	−	0.882843i	$-0.655626\pi$
$17$	−3.87298	−0.939336	−0.469668	+	0.882843i	$0.655626\pi$
$18$	0	0
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	6.70820i	1.50000i
$21$	0	0
$22$	10.0000	2.13201
$23$	−7.74597	−1.61515	−0.807573	−	0.589768i	$-0.799219\pi$
$23$	−7.74597	−1.61515	−0.807573	+	0.589768i	$0.799219\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	10.3923i	1.96396i
$29$	3.87298	0.719195	0.359597	−	0.933108i	$-0.382914\pi$
$29$	3.87298	0.719195	0.359597	+	0.933108i	$0.382914\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	6.70820i	1.18585i
$33$	0	0
$34$	8.66025i	1.48522i
$35$	−7.74597	−1.30931
$36$	0	0
$37$	− 1.73205i	− 0.284747i	−0.989813	−	0.142374i	$-0.954527\pi$
$37$	− 1.73205i	− 0.284747i	0.989813	−	0.142374i	$-0.0454735\pi$
$38$	−7.74597	−1.25656
$39$	0	0
$40$	5.00000	0.790569
$41$	− 2.23607i	− 0.349215i	−0.984638	−	0.174608i	$-0.944134\pi$
$41$	− 2.23607i	− 0.349215i	0.984638	−	0.174608i	$-0.0558657\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	− 13.4164i	− 2.02260i
$45$	0	0
$46$	17.3205i	2.55377i
$47$	4.47214i	0.652328i	0.945313	+	0.326164i	$0.105756\pi$
$47$	4.47214i	0.652328i	−0.945313	+	0.326164i	$0.894244\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.6190	−1.59599	−0.797993	−	0.602667i	$-0.794105\pi$
$53$	−11.6190	−1.59599	−0.797993	+	0.602667i	$0.794105\pi$
$54$	0	0
$55$	10.0000	1.34840
$56$	7.74597	1.03510
$57$	0	0
$58$	− 8.66025i	− 1.13715i
$59$	− 8.94427i	− 1.16445i	−0.813029	−	0.582223i	$-0.802183\pi$
$59$	− 8.94427i	− 1.16445i	0.813029	−	0.582223i	$-0.197817\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	0	0
$63$	0	0
$64$	13.0000	1.62500
$65$	0	0
$66$	0	0
$67$	3.46410i	0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$	3.46410i	0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$	11.6190	1.40900
$69$	0	0
$70$	17.3205i	2.07020i
$71$	4.47214i	0.530745i	0.964146	+	0.265372i	$0.0854949\pi$
$71$	4.47214i	0.530745i	−0.964146	+	0.265372i	$0.914505\pi$
$72$	0	0
$73$	− 15.5885i	− 1.82449i	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	− 15.5885i	− 1.82449i	0.409644	−	0.912245i	$-0.365653\pi$
$74$	−3.87298	−0.450225
$75$	0	0
$76$	10.3923i	1.19208i
$77$	15.4919	1.76547
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	2.23607i	0.250000i
$81$	0	0
$82$	−5.00000	−0.552158
$83$	4.47214i	0.490881i	0.969412	+	0.245440i	$0.0789325\pi$
$83$	4.47214i	0.490881i	−0.969412	+	0.245440i	$0.921067\pi$
$84$	0	0
$85$	8.66025i	0.939336i
$86$	− 4.47214i	− 0.482243i
$87$	0	0
$88$	−10.0000	−1.06600
$89$	4.47214i	0.474045i	0.971504	+	0.237023i	$0.0761716\pi$
$89$	4.47214i	0.474045i	−0.971504	+	0.237023i	$0.923828\pi$
$90$	0	0
$91$	0	0
$92$	23.2379	2.42272
$93$	0	0
$94$	10.0000	1.03142
$95$	−7.74597	−0.794719
$96$	0	0
$97$	6.92820i	0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$	6.92820i	0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$	11.1803i	1.12938i
$99$	0	0
$100$	0	0
$101$	3.87298	0.385376	0.192688	−	0.981260i	$-0.438279\pi$
$101$	3.87298	0.385376	0.192688	+	0.981260i	$0.438279\pi$
$102$	0	0
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	0	0
$105$	0	0
$106$	25.9808i	2.52347i
$107$	−7.74597	−0.748831	−0.374415	−	0.927261i	$-0.622157\pi$
$107$	−7.74597	−0.748831	−0.374415	+	0.927261i	$0.622157\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	− 22.3607i	− 2.13201i
$111$	0	0
$112$	3.46410i	0.327327i
$113$	19.3649	1.82170	0.910849	−	0.412740i	$-0.135428\pi$
$113$	19.3649	1.82170	0.910849	+	0.412740i	$0.135428\pi$
$114$	0	0
$115$	17.3205i	1.61515i
$116$	−11.6190	−1.07879
$117$	0	0
$118$	−20.0000	−1.84115
$119$	13.4164i	1.22988i
$120$	0	0
$121$	−9.00000	−0.818182
$122$	15.6525i	1.41711i
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	− 15.6525i	− 1.38350i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	7.74597	0.669150
$135$	0	0
$136$	− 8.66025i	− 0.742611i
$137$	− 15.6525i	− 1.33728i	−0.743586	−	0.668641i	$-0.766876\pi$
$137$	− 15.6525i	− 1.33728i	0.743586	−	0.668641i	$-0.233124\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	23.2379	1.96396
$141$	0	0
$142$	10.0000	0.839181
$143$	0	0
$144$	0	0
$145$	− 8.66025i	− 0.719195i
$146$	−34.8569	−2.88477
$147$	0	0
$148$	5.19615i	0.427121i
$149$	11.1803i	0.915929i	0.888970	+	0.457965i	$0.151421\pi$
$149$	11.1803i	0.915929i	−0.888970	+	0.457965i	$0.848579\pi$
$150$	0	0
$151$	10.3923i	0.845714i	0.906196	+	0.422857i	$0.138973\pi$
$151$	10.3923i	0.845714i	−0.906196	+	0.422857i	$0.861027\pi$
$152$	7.74597	0.628281
$153$	0	0
$154$	− 34.6410i	− 2.79145i
$155$	0	0
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	− 17.8885i	− 1.42314i
$159$	0	0
$160$	15.0000	1.18585
$161$	26.8328i	2.11472i
$162$	0	0
$163$	− 13.8564i	− 1.08532i	−0.839953	−	0.542659i	$-0.817418\pi$
$163$	− 13.8564i	− 1.08532i	0.839953	−	0.542659i	$-0.182582\pi$
$164$	6.70820i	0.523823i
$165$	0	0
$166$	10.0000	0.776151
$167$	− 8.94427i	− 0.692129i	−0.938211	−	0.346064i	$-0.887518\pi$
$167$	− 8.94427i	− 0.692129i	0.938211	−	0.346064i	$-0.112482\pi$
$168$	0	0
$169$	0	0
$170$	19.3649	1.48522
$171$	0	0
$172$	−6.00000	−0.457496
$173$	−15.4919	−1.17783	−0.588915	−	0.808195i	$-0.700445\pi$
$173$	−15.4919	−1.17783	−0.588915	+	0.808195i	$0.700445\pi$
$174$	0	0
$175$	0	0
$176$	− 4.47214i	− 0.337100i
$177$	0	0
$178$	10.0000	0.749532
$179$	−7.74597	−0.578961	−0.289480	−	0.957184i	$-0.593482\pi$
$179$	−7.74597	−0.578961	−0.289480	+	0.957184i	$0.593482\pi$
$180$	0	0
$181$	−19.0000	−1.41226	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000	−1.41226	−0.706129	+	0.708083i	$0.749560\pi$
$182$	0	0
$183$	0	0
$184$	− 17.3205i	− 1.27688i
$185$	−3.87298	−0.284747
$186$	0	0
$187$	− 17.3205i	− 1.26660i
$188$	− 13.4164i	− 0.978492i
$189$	0	0
$190$	17.3205i	1.25656i
$191$	−15.4919	−1.12096	−0.560478	−	0.828169i	$-0.689383\pi$
$191$	−15.4919	−1.12096	−0.560478	+	0.828169i	$0.689383\pi$
$192$	0	0
$193$	− 1.73205i	− 0.124676i	−0.998055	−	0.0623379i	$-0.980144\pi$
$193$	− 1.73205i	− 0.124676i	0.998055	−	0.0623379i	$-0.0198556\pi$
$194$	15.4919	1.11226
$195$	0	0
$196$	15.0000	1.07143
$197$	4.47214i	0.318626i	0.987228	+	0.159313i	$0.0509280\pi$
$197$	4.47214i	0.318626i	−0.987228	+	0.159313i	$0.949072\pi$
$198$	0	0
$199$	−22.0000	−1.55954	−0.779769	−	0.626067i	$-0.784664\pi$
$199$	−22.0000	−1.55954	−0.779769	+	0.626067i	$0.784664\pi$
$200$	0	0
$201$	0	0
$202$	− 8.66025i	− 0.609333i
$203$	− 13.4164i	− 0.941647i
$204$	0	0
$205$	−5.00000	−0.349215
$206$	− 4.47214i	− 0.311588i
$207$	0	0
$208$	0	0
$209$	15.4919	1.07160
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	34.8569	2.39398
$213$	0	0
$214$	17.3205i	1.18401i
$215$	− 4.47214i	− 0.304997i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−30.0000	−2.02260
$221$	0	0
$222$	0	0
$223$	− 27.7128i	− 1.85579i	−0.372845	−	0.927894i	$-0.621618\pi$
$223$	− 27.7128i	− 1.85579i	0.372845	−	0.927894i	$-0.378382\pi$
$224$	23.2379	1.55265
$225$	0	0
$226$	− 43.3013i	− 2.88036i
$227$	− 22.3607i	− 1.48413i	−0.670328	−	0.742065i	$-0.733846\pi$
$227$	− 22.3607i	− 1.48413i	0.670328	−	0.742065i	$-0.266154\pi$
$228$	0	0
$229$	− 20.7846i	− 1.37349i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	− 20.7846i	− 1.37349i	0.726900	−	0.686743i	$-0.240960\pi$
$230$	38.7298	2.55377
$231$	0	0
$232$	8.66025i	0.568574i
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	10.0000	0.652328
$236$	26.8328i	1.74667i
$237$	0	0
$238$	30.0000	1.94461
$239$	4.47214i	0.289278i	0.989484	+	0.144639i	$0.0462022\pi$
$239$	4.47214i	0.289278i	−0.989484	+	0.144639i	$0.953798\pi$
$240$	0	0
$241$	− 19.0526i	− 1.22728i	−0.789585	−	0.613642i	$-0.789704\pi$
$241$	− 19.0526i	− 1.22728i	0.789585	−	0.613642i	$-0.210296\pi$
$242$	20.1246i	1.29366i
$243$	0	0
$244$	21.0000	1.34439
$245$	11.1803i	0.714286i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−25.0000	−1.58114
$251$	−15.4919	−0.977842	−0.488921	−	0.872328i	$-0.662609\pi$
$251$	−15.4919	−0.977842	−0.488921	+	0.872328i	$0.662609\pi$
$252$	0	0
$253$	− 34.6410i	− 2.17786i
$254$	8.94427i	0.561214i
$255$	0	0
$256$	−9.00000	−0.562500
$257$	3.87298	0.241590	0.120795	−	0.992677i	$-0.461456\pi$
$257$	3.87298	0.241590	0.120795	+	0.992677i	$0.461456\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.74597	−0.477637	−0.238818	−	0.971064i	$-0.576760\pi$
$263$	−7.74597	−0.477637	−0.238818	+	0.971064i	$0.576760\pi$
$264$	0	0
$265$	25.9808i	1.59599i
$266$	26.8328i	1.64523i
$267$	0	0
$268$	− 10.3923i	− 0.634811i
$269$	−15.4919	−0.944560	−0.472280	−	0.881449i	$-0.656569\pi$
$269$	−15.4919	−0.944560	−0.472280	+	0.881449i	$0.656569\pi$
$270$	0	0
$271$	− 6.92820i	− 0.420858i	−0.977609	−	0.210429i	$-0.932514\pi$
$271$	− 6.92820i	− 0.420858i	0.977609	−	0.210429i	$-0.0674861\pi$
$272$	3.87298	0.234834
$273$	0	0
$274$	−35.0000	−2.11443
$275$	0	0
$276$	0	0
$277$	11.0000	0.660926	0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$278$	− 17.8885i	− 1.07288i
$279$	0	0
$280$	− 17.3205i	− 1.03510i
$281$	− 15.6525i	− 0.933748i	−0.884324	−	0.466874i	$-0.845380\pi$
$281$	− 15.6525i	− 0.933748i	0.884324	−	0.466874i	$-0.154620\pi$
$282$	0	0
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	− 13.4164i	− 0.796117i
$285$	0	0
$286$	0	0
$287$	−7.74597	−0.457230
$288$	0	0
$289$	−2.00000	−0.117647
$290$	−19.3649	−1.13715
$291$	0	0
$292$	46.7654i	2.73674i
$293$	− 2.23607i	− 0.130632i	−0.997865	−	0.0653162i	$-0.979194\pi$
$293$	− 2.23607i	− 0.130632i	0.997865	−	0.0653162i	$-0.0208056\pi$
$294$	0	0
$295$	−20.0000	−1.16445
$296$	3.87298	0.225113
$297$	0	0
$298$	25.0000	1.44821
$299$	0	0
$300$	0	0
$301$	− 6.92820i	− 0.399335i
$302$	23.2379	1.33719
$303$	0	0
$304$	3.46410i	0.198680i
$305$	15.6525i	0.896258i
$306$	0	0
$307$	− 10.3923i	− 0.593120i	−0.955014	−	0.296560i	$-0.904160\pi$
$307$	− 10.3923i	− 0.593120i	0.955014	−	0.296560i	$-0.0958395\pi$
$308$	−46.4758	−2.64820
$309$	0	0
$310$	0	0
$311$	−23.2379	−1.31770	−0.658850	−	0.752274i	$-0.728957\pi$
$311$	−23.2379	−1.31770	−0.658850	+	0.752274i	$0.728957\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	− 24.5967i	− 1.38807i
$315$	0	0
$316$	−24.0000	−1.35011
$317$	− 29.0689i	− 1.63267i	−0.577578	−	0.816336i	$-0.696002\pi$
$317$	− 29.0689i	− 1.63267i	0.577578	−	0.816336i	$-0.303998\pi$
$318$	0	0
$319$	17.3205i	0.969762i
$320$	− 29.0689i	− 1.62500i
$321$	0	0
$322$	60.0000	3.34367
$323$	13.4164i	0.746509i
$324$	0	0
$325$	0	0
$326$	−30.9839	−1.71604
$327$	0	0
$328$	5.00000	0.276079
$329$	15.4919	0.854098
$330$	0	0
$331$	27.7128i	1.52323i	0.648027	+	0.761617i	$0.275594\pi$
$331$	27.7128i	1.52323i	−0.648027	+	0.761617i	$0.724406\pi$
$332$	− 13.4164i	− 0.736321i
$333$	0	0
$334$	−20.0000	−1.09435
$335$	7.74597	0.423207
$336$	0	0
$337$	11.0000	0.599208	0.299604	−	0.954064i	$-0.403145\pi$
$337$	11.0000	0.599208	0.299604	+	0.954064i	$0.403145\pi$
$338$	0	0
$339$	0	0
$340$	− 25.9808i	− 1.40900i
$341$	0	0
$342$	0	0
$343$	− 6.92820i	− 0.374088i
$344$	4.47214i	0.241121i
$345$	0	0
$346$	34.6410i	1.86231i
$347$	7.74597	0.415825	0.207913	−	0.978147i	$-0.433333\pi$
$347$	7.74597	0.415825	0.207913	+	0.978147i	$0.433333\pi$
$348$	0	0
$349$	13.8564i	0.741716i	0.928689	+	0.370858i	$0.120936\pi$
$349$	13.8564i	0.741716i	−0.928689	+	0.370858i	$0.879064\pi$
$350$	0	0
$351$	0	0
$352$	−30.0000	−1.59901
$353$	− 2.23607i	− 0.119014i	−0.998228	−	0.0595069i	$-0.981047\pi$
$353$	− 2.23607i	− 0.119014i	0.998228	−	0.0595069i	$-0.0189528\pi$
$354$	0	0
$355$	10.0000	0.530745
$356$	− 13.4164i	− 0.711068i
$357$	0	0
$358$	17.3205i	0.915417i
$359$	31.3050i	1.65221i	0.563515	+	0.826106i	$0.309449\pi$
$359$	31.3050i	1.65221i	−0.563515	+	0.826106i	$0.690551\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	42.4853i	2.23298i
$363$	0	0
$364$	0	0
$365$	−34.8569	−1.82449
$366$	0	0
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	7.74597	0.403786
$369$	0	0
$370$	8.66025i	0.450225i
$371$	40.2492i	2.08964i
$372$	0	0
$373$	−37.0000	−1.91579	−0.957894	−	0.287123i	$-0.907301\pi$
$373$	−37.0000	−1.91579	−0.957894	+	0.287123i	$0.907301\pi$
$374$	−38.7298	−2.00267
$375$	0	0
$376$	−10.0000	−0.515711
$377$	0	0
$378$	0	0
$379$	13.8564i	0.711756i	0.934532	+	0.355878i	$0.115818\pi$
$379$	13.8564i	0.711756i	−0.934532	+	0.355878i	$0.884182\pi$
$380$	23.2379	1.19208
$381$	0	0
$382$	34.6410i	1.77239i
$383$	17.8885i	0.914062i	0.889451	+	0.457031i	$0.151087\pi$
$383$	17.8885i	0.914062i	−0.889451	+	0.457031i	$0.848913\pi$
$384$	0	0
$385$	− 34.6410i	− 1.76547i
$386$	−3.87298	−0.197130
$387$	0	0
$388$	− 20.7846i	− 1.05518i
$389$	−11.6190	−0.589104	−0.294552	−	0.955635i	$-0.595170\pi$
$389$	−11.6190	−0.589104	−0.294552	+	0.955635i	$0.595170\pi$
$390$	0	0
$391$	30.0000	1.51717
$392$	− 11.1803i	− 0.564692i
$393$	0	0
$394$	10.0000	0.503793
$395$	− 17.8885i	− 0.900070i
$396$	0	0
$397$	− 13.8564i	− 0.695433i	−0.937600	−	0.347717i	$-0.886957\pi$
$397$	− 13.8564i	− 0.695433i	0.937600	−	0.347717i	$-0.113043\pi$
$398$	49.1935i	2.46585i
$399$	0	0
$400$	0	0
$401$	− 29.0689i	− 1.45163i	−0.687890	−	0.725815i	$-0.741463\pi$
$401$	− 29.0689i	− 1.45163i	0.687890	−	0.725815i	$-0.258537\pi$
$402$	0	0
$403$	0	0
$404$	−11.6190	−0.578064
$405$	0	0
$406$	−30.0000	−1.48888
$407$	7.74597	0.383953
$408$	0	0
$409$	− 8.66025i	− 0.428222i	−0.976809	−	0.214111i	$-0.931315\pi$
$409$	− 8.66025i	− 0.428222i	0.976809	−	0.214111i	$-0.0686854\pi$
$410$	11.1803i	0.552158i
$411$	0	0
$412$	−6.00000	−0.295599
$413$	−30.9839	−1.52462
$414$	0	0
$415$	10.0000	0.490881
$416$	0	0
$417$	0	0
$418$	− 34.6410i	− 1.69435i
$419$	15.4919	0.756830	0.378415	−	0.925636i	$-0.376469\pi$
$419$	15.4919	0.756830	0.378415	+	0.925636i	$0.376469\pi$
$420$	0	0
$421$	− 15.5885i	− 0.759735i	−0.925041	−	0.379867i	$-0.875970\pi$
$421$	− 15.5885i	− 0.759735i	0.925041	−	0.379867i	$-0.124030\pi$
$422$	− 17.8885i	− 0.870801i
$423$	0	0
$424$	− 25.9808i	− 1.26174i
$425$	0	0
$426$	0	0
$427$	24.2487i	1.17348i
$428$	23.2379	1.12325
$429$	0	0
$430$	−10.0000	−0.482243
$431$	4.47214i	0.215415i	0.994183	+	0.107708i	$0.0343510\pi$
$431$	4.47214i	0.215415i	−0.994183	+	0.107708i	$0.965649\pi$
$432$	0	0
$433$	35.0000	1.68199	0.840996	−	0.541041i	$-0.181970\pi$
$433$	35.0000	1.68199	0.840996	+	0.541041i	$0.181970\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.8328i	1.28359i
$438$	0	0
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	22.3607i	1.06600i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	−61.9677	−2.93426
$447$	0	0
$448$	− 45.0333i	− 2.12762i
$449$	4.47214i	0.211053i	0.994416	+	0.105527i	$0.0336528\pi$
$449$	4.47214i	0.211053i	−0.994416	+	0.105527i	$0.966347\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	−58.0948	−2.73255
$453$	0	0
$454$	−50.0000	−2.34662
$455$	0	0
$456$	0	0
$457$	− 12.1244i	− 0.567153i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	− 12.1244i	− 0.567153i	0.958950	−	0.283577i	$-0.0915211\pi$
$458$	−46.4758	−2.17167
$459$	0	0
$460$	− 51.9615i	− 2.42272i
$461$	− 2.23607i	− 0.104144i	−0.998643	−	0.0520720i	$-0.983417\pi$
$461$	− 2.23607i	− 0.104144i	0.998643	−	0.0520720i	$-0.0165825\pi$
$462$	0	0
$463$	− 10.3923i	− 0.482971i	−0.970404	−	0.241486i	$-0.922365\pi$
$463$	− 10.3923i	− 0.482971i	0.970404	−	0.241486i	$-0.0776347\pi$
$464$	−3.87298	−0.179799
$465$	0	0
$466$	0	0
$467$	23.2379	1.07532	0.537661	−	0.843161i	$-0.319308\pi$
$467$	23.2379	1.07532	0.537661	+	0.843161i	$0.319308\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	− 22.3607i	− 1.03142i
$471$	0	0
$472$	20.0000	0.920575
$473$	8.94427i	0.411258i
$474$	0	0
$475$	0	0
$476$	− 40.2492i	− 1.84482i
$477$	0	0
$478$	10.0000	0.457389
$479$	17.8885i	0.817348i	0.912680	+	0.408674i	$0.134009\pi$
$479$	17.8885i	0.817348i	−0.912680	+	0.408674i	$0.865991\pi$
$480$	0	0
$481$	0	0
$482$	−42.6028	−1.94051
$483$	0	0
$484$	27.0000	1.22727
$485$	15.4919	0.703452
$486$	0	0
$487$	38.1051i	1.72671i	0.504599	+	0.863354i	$0.331640\pi$
$487$	38.1051i	1.72671i	−0.504599	+	0.863354i	$0.668360\pi$
$488$	− 15.6525i	− 0.708554i
$489$	0	0
$490$	25.0000	1.12938
$491$	38.7298	1.74785	0.873926	−	0.486058i	$-0.161566\pi$
$491$	38.7298	1.74785	0.873926	+	0.486058i	$0.161566\pi$
$492$	0	0
$493$	−15.0000	−0.675566
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.4919	0.694908
$498$	0	0
$499$	− 20.7846i	− 0.930447i	−0.885193	−	0.465223i	$-0.845974\pi$
$499$	− 20.7846i	− 0.930447i	0.885193	−	0.465223i	$-0.154026\pi$
$500$	33.5410i	1.50000i
$501$	0	0
$502$	34.6410i	1.54610i
$503$	7.74597	0.345376	0.172688	−	0.984977i	$-0.444755\pi$
$503$	7.74597	0.345376	0.172688	+	0.984977i	$0.444755\pi$
$504$	0	0
$505$	− 8.66025i	− 0.385376i
$506$	−77.4597	−3.44350
$507$	0	0
$508$	12.0000	0.532414
$509$	− 29.0689i	− 1.28846i	−0.764834	−	0.644228i	$-0.777179\pi$
$509$	− 29.0689i	− 1.28846i	0.764834	−	0.644228i	$-0.222821\pi$
$510$	0	0
$511$	−54.0000	−2.38882
$512$	− 11.1803i	− 0.494106i
$513$	0	0
$514$	− 8.66025i	− 0.381987i
$515$	− 4.47214i	− 0.197066i
$516$	0	0
$517$	−20.0000	−0.879599
$518$	13.4164i	0.589483i
$519$	0	0
$520$	0	0
$521$	11.6190	0.509035	0.254518	−	0.967068i	$-0.418083\pi$
$521$	11.6190	0.509035	0.254518	+	0.967068i	$0.418083\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	0	0
$525$	0	0
$526$	17.3205i	0.755210i
$527$	0	0
$528$	0	0
$529$	37.0000	1.60870
$530$	58.0948	2.52347
$531$	0	0
$532$	36.0000	1.56080
$533$	0	0
$534$	0	0
$535$	17.3205i	0.748831i
$536$	−7.74597	−0.334575
$537$	0	0
$538$	34.6410i	1.49348i
$539$	− 22.3607i	− 0.963143i
$540$	0	0
$541$	25.9808i	1.11700i	0.829504	+	0.558500i	$0.188623\pi$
$541$	25.9808i	1.11700i	−0.829504	+	0.558500i	$0.811377\pi$
$542$	−15.4919	−0.665436
$543$	0	0
$544$	− 25.9808i	− 1.11392i
$545$	0	0
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	46.9574i	2.00592i
$549$	0	0
$550$	0	0
$551$	− 13.4164i	− 0.571558i
$552$	0	0
$553$	− 27.7128i	− 1.17847i
$554$	− 24.5967i	− 1.04502i
$555$	0	0
$556$	−24.0000	−1.01783
$557$	− 29.0689i	− 1.23169i	−0.787868	−	0.615844i	$-0.788815\pi$
$557$	− 29.0689i	− 1.23169i	0.787868	−	0.615844i	$-0.211185\pi$
$558$	0	0
$559$	0	0
$560$	7.74597	0.327327
$561$	0	0
$562$	−35.0000	−1.47639
$563$	30.9839	1.30581	0.652907	−	0.757438i	$-0.273549\pi$
$563$	30.9839	1.30581	0.652907	+	0.757438i	$0.273549\pi$
$564$	0	0
$565$	− 43.3013i	− 1.82170i
$566$	22.3607i	0.939889i
$567$	0	0
$568$	−10.0000	−0.419591
$569$	15.4919	0.649456	0.324728	−	0.945808i	$-0.394727\pi$
$569$	15.4919	0.649456	0.324728	+	0.945808i	$0.394727\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	0	0
$573$	0	0
$574$	17.3205i	0.722944i
$575$	0	0
$576$	0	0
$577$	− 5.19615i	− 0.216319i	−0.994134	−	0.108159i	$-0.965504\pi$
$577$	− 5.19615i	− 0.216319i	0.994134	−	0.108159i	$-0.0344957\pi$
$578$	4.47214i	0.186016i
$579$	0	0
$580$	25.9808i	1.07879i
$581$	15.4919	0.642714
$582$	0	0
$583$	− 51.9615i	− 2.15203i
$584$	34.8569	1.44239
$585$	0	0
$586$	−5.00000	−0.206548
$587$	− 35.7771i	− 1.47668i	−0.674430	−	0.738339i	$-0.735610\pi$
$587$	− 35.7771i	− 1.47668i	0.674430	−	0.738339i	$-0.264390\pi$
$588$	0	0
$589$	0	0
$590$	44.7214i	1.84115i
$591$	0	0
$592$	1.73205i	0.0711868i
$593$	− 2.23607i	− 0.0918243i	−0.998945	−	0.0459122i	$-0.985381\pi$
$593$	− 2.23607i	− 0.0918243i	0.998945	−	0.0459122i	$-0.0146194\pi$
$594$	0	0
$595$	30.0000	1.22988
$596$	− 33.5410i	− 1.37389i
$597$	0	0
$598$	0	0
$599$	46.4758	1.89895	0.949475	−	0.313843i	$-0.101617\pi$
$599$	46.4758	1.89895	0.949475	+	0.313843i	$0.101617\pi$
$600$	0	0
$601$	29.0000	1.18293	0.591467	−	0.806329i	$-0.298549\pi$
$601$	29.0000	1.18293	0.591467	+	0.806329i	$0.298549\pi$
$602$	−15.4919	−0.631404
$603$	0	0
$604$	− 31.1769i	− 1.26857i
$605$	20.1246i	0.818182i
$606$	0	0
$607$	20.0000	0.811775	0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000	0.811775	0.405887	+	0.913923i	$0.366962\pi$
$608$	23.2379	0.942421
$609$	0	0
$610$	35.0000	1.41711
$611$	0	0
$612$	0	0
$613$	− 1.73205i	− 0.0699569i	−0.999388	−	0.0349784i	$-0.988864\pi$
$613$	− 1.73205i	− 0.0699569i	0.999388	−	0.0349784i	$-0.0111363\pi$
$614$	−23.2379	−0.937805
$615$	0	0
$616$	34.6410i	1.39573i
$617$	− 29.0689i	− 1.17027i	−0.810936	−	0.585135i	$-0.801042\pi$
$617$	− 29.0689i	− 1.17027i	0.810936	−	0.585135i	$-0.198958\pi$
$618$	0	0
$619$	20.7846i	0.835404i	0.908584	+	0.417702i	$0.137164\pi$
$619$	20.7846i	0.835404i	−0.908584	+	0.417702i	$0.862836\pi$
$620$	0	0
$621$	0	0
$622$	51.9615i	2.08347i
$623$	15.4919	0.620671
$624$	0	0
$625$	−25.0000	−1.00000
$626$	49.1935i	1.96617i
$627$	0	0
$628$	−33.0000	−1.31684
$629$	6.70820i	0.267474i
$630$	0	0
$631$	6.92820i	0.275807i	0.990446	+	0.137904i	$0.0440364\pi$
$631$	6.92820i	0.275807i	−0.990446	+	0.137904i	$0.955964\pi$
$632$	17.8885i	0.711568i
$633$	0	0
$634$	−65.0000	−2.58148
$635$	8.94427i	0.354943i
$636$	0	0
$637$	0	0
$638$	38.7298	1.53333
$639$	0	0
$640$	−35.0000	−1.38350
$641$	19.3649	0.764868	0.382434	−	0.923983i	$-0.375086\pi$
$641$	19.3649	0.764868	0.382434	+	0.923983i	$0.375086\pi$
$642$	0	0
$643$	27.7128i	1.09289i	0.837496	+	0.546443i	$0.184019\pi$
$643$	27.7128i	1.09289i	−0.837496	+	0.546443i	$0.815981\pi$
$644$	− 80.4984i	− 3.17208i
$645$	0	0
$646$	30.0000	1.18033
$647$	−30.9839	−1.21810	−0.609051	−	0.793131i	$-0.708449\pi$
$647$	−30.9839	−1.21810	−0.609051	+	0.793131i	$0.708449\pi$
$648$	0	0
$649$	40.0000	1.57014
$650$	0	0
$651$	0	0
$652$	41.5692i	1.62798i
$653$	15.4919	0.606246	0.303123	−	0.952951i	$-0.401971\pi$
$653$	15.4919	0.606246	0.303123	+	0.952951i	$0.401971\pi$
$654$	0	0
$655$	0	0
$656$	2.23607i	0.0873038i
$657$	0	0
$658$	− 34.6410i	− 1.35045i
$659$	30.9839	1.20696	0.603480	−	0.797378i	$-0.293780\pi$
$659$	30.9839	1.20696	0.603480	+	0.797378i	$0.293780\pi$
$660$	0	0
$661$	19.0526i	0.741059i	0.928821	+	0.370529i	$0.120824\pi$
$661$	19.0526i	0.741059i	−0.928821	+	0.370529i	$0.879176\pi$
$662$	61.9677	2.40844
$663$	0	0
$664$	−10.0000	−0.388075
$665$	26.8328i	1.04053i
$666$	0	0
$667$	−30.0000	−1.16160
$668$	26.8328i	1.03819i
$669$	0	0
$670$	− 17.3205i	− 0.669150i
$671$	− 31.3050i	− 1.20851i
$672$	0	0
$673$	−43.0000	−1.65753	−0.828764	−	0.559598i	$-0.810955\pi$
$673$	−43.0000	−1.65753	−0.828764	+	0.559598i	$0.810955\pi$
$674$	− 24.5967i	− 0.947431i
$675$	0	0
$676$	0	0
$677$	46.4758	1.78621	0.893105	−	0.449848i	$-0.148522\pi$
$677$	46.4758	1.78621	0.893105	+	0.449848i	$0.148522\pi$
$678$	0	0
$679$	24.0000	0.921035
$680$	−19.3649	−0.742611
$681$	0	0
$682$	0	0
$683$	− 35.7771i	− 1.36897i	−0.729026	−	0.684486i	$-0.760027\pi$
$683$	− 35.7771i	− 1.36897i	0.729026	−	0.684486i	$-0.239973\pi$
$684$	0	0
$685$	−35.0000	−1.33728
$686$	−15.4919	−0.591485
$687$	0	0
$688$	−2.00000	−0.0762493
$689$	0	0
$690$	0	0
$691$	24.2487i	0.922464i	0.887279	+	0.461232i	$0.152592\pi$
$691$	24.2487i	0.922464i	−0.887279	+	0.461232i	$0.847408\pi$
$692$	46.4758	1.76674
$693$	0	0
$694$	− 17.3205i	− 0.657477i
$695$	− 17.8885i	− 0.678551i
$696$	0	0
$697$	8.66025i	0.328031i
$698$	30.9839	1.17276
$699$	0	0
$700$	0	0
$701$	−46.4758	−1.75537	−0.877683	−	0.479241i	$-0.840912\pi$
$701$	−46.4758	−1.75537	−0.877683	+	0.479241i	$0.840912\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	58.1378i	2.19115i
$705$	0	0
$706$	−5.00000	−0.188177
$707$	− 13.4164i	− 0.504576i
$708$	0	0
$709$	12.1244i	0.455340i	0.973738	+	0.227670i	$0.0731107\pi$
$709$	12.1244i	0.455340i	−0.973738	+	0.227670i	$0.926889\pi$
$710$	− 22.3607i	− 0.839181i
$711$	0	0
$712$	−10.0000	−0.374766
$713$	0	0
$714$	0	0
$715$	0	0
$716$	23.2379	0.868441
$717$	0	0
$718$	70.0000	2.61238
$719$	30.9839	1.15550	0.577752	−	0.816213i	$-0.303930\pi$
$719$	30.9839	1.15550	0.577752	+	0.816213i	$0.303930\pi$
$720$	0	0
$721$	− 6.92820i	− 0.258020i
$722$	− 15.6525i	− 0.582525i
$723$	0	0
$724$	57.0000	2.11839
$725$	0	0
$726$	0	0
$727$	−46.0000	−1.70605	−0.853023	−	0.521874i	$-0.825233\pi$
$727$	−46.0000	−1.70605	−0.853023	+	0.521874i	$0.825233\pi$
$728$	0	0
$729$	0	0
$730$	77.9423i	2.88477i
$731$	−7.74597	−0.286495
$732$	0	0
$733$	36.3731i	1.34347i	0.740792	+	0.671735i	$0.234451\pi$
$733$	36.3731i	1.34347i	−0.740792	+	0.671735i	$0.765549\pi$
$734$	− 31.3050i	− 1.15549i
$735$	0	0
$736$	− 51.9615i	− 1.91533i
$737$	−15.4919	−0.570653
$738$	0	0
$739$	34.6410i	1.27429i	0.770744	+	0.637145i	$0.219885\pi$
$739$	34.6410i	1.27429i	−0.770744	+	0.637145i	$0.780115\pi$
$740$	11.6190	0.427121
$741$	0	0
$742$	90.0000	3.30400
$743$	− 8.94427i	− 0.328134i	−0.986449	−	0.164067i	$-0.947539\pi$
$743$	− 8.94427i	− 0.328134i	0.986449	−	0.164067i	$-0.0524612\pi$
$744$	0	0
$745$	25.0000	0.915929
$746$	82.7345i	3.02913i
$747$	0	0
$748$	51.9615i	1.89990i
$749$	26.8328i	0.980450i
$750$	0	0
$751$	38.0000	1.38664	0.693320	−	0.720630i	$-0.256147\pi$
$751$	38.0000	1.38664	0.693320	+	0.720630i	$0.256147\pi$
$752$	− 4.47214i	− 0.163082i
$753$	0	0
$754$	0	0
$755$	23.2379	0.845714
$756$	0	0
$757$	−46.0000	−1.67190	−0.835949	−	0.548807i	$-0.815082\pi$
$757$	−46.0000	−1.67190	−0.835949	+	0.548807i	$0.815082\pi$
$758$	30.9839	1.12538
$759$	0	0
$760$	− 17.3205i	− 0.628281i
$761$	31.3050i	1.13480i	0.823441	+	0.567402i	$0.192051\pi$
$761$	31.3050i	1.13480i	−0.823441	+	0.567402i	$0.807949\pi$
$762$	0	0
$763$	0	0
$764$	46.4758	1.68144
$765$	0	0
$766$	40.0000	1.44526
$767$	0	0
$768$	0	0
$769$	− 48.4974i	− 1.74886i	−0.485150	−	0.874431i	$-0.661235\pi$
$769$	− 48.4974i	− 1.74886i	0.485150	−	0.874431i	$-0.338765\pi$
$770$	−77.4597	−2.79145
$771$	0	0
$772$	5.19615i	0.187014i
$773$	4.47214i	0.160852i	0.996761	+	0.0804258i	$0.0256280\pi$
$773$	4.47214i	0.160852i	−0.996761	+	0.0804258i	$0.974372\pi$
$774$	0	0
$775$	0	0
$776$	−15.4919	−0.556128
$777$	0	0
$778$	25.9808i	0.931455i
$779$	−7.74597	−0.277528
$780$	0	0
$781$	−20.0000	−0.715656
$782$	− 67.0820i	− 2.39885i
$783$	0	0
$784$	5.00000	0.178571
$785$	− 24.5967i	− 0.877896i
$786$	0	0
$787$	27.7128i	0.987855i	0.869503	+	0.493928i	$0.164439\pi$
$787$	27.7128i	0.987855i	−0.869503	+	0.493928i	$0.835561\pi$
$788$	− 13.4164i	− 0.477940i
$789$	0	0
$790$	−40.0000	−1.42314
$791$	− 67.0820i	− 2.38516i
$792$	0	0
$793$	0	0
$794$	−30.9839	−1.09958
$795$	0	0
$796$	66.0000	2.33931
$797$	30.9839	1.09750	0.548752	−	0.835985i	$-0.315103\pi$
$797$	30.9839	1.09750	0.548752	+	0.835985i	$0.315103\pi$
$798$	0	0
$799$	− 17.3205i	− 0.612756i
$800$	0	0
$801$	0	0
$802$	−65.0000	−2.29523
$803$	69.7137	2.46014
$804$	0	0
$805$	60.0000	2.11472
$806$	0	0
$807$	0	0
$808$	8.66025i	0.304667i
$809$	−42.6028	−1.49784	−0.748918	−	0.662663i	$-0.769426\pi$
$809$	−42.6028	−1.49784	−0.748918	+	0.662663i	$0.769426\pi$
$810$	0	0
$811$	41.5692i	1.45969i	0.683611	+	0.729846i	$0.260408\pi$
$811$	41.5692i	1.45969i	−0.683611	+	0.729846i	$0.739592\pi$
$812$	40.2492i	1.41247i
$813$	0	0
$814$	− 17.3205i	− 0.607083i
$815$	−30.9839	−1.08532
$816$	0	0
$817$	− 6.92820i	− 0.242387i
$818$	−19.3649	−0.677078
$819$	0	0
$820$	15.0000	0.523823
$821$	4.47214i	0.156079i	0.996950	+	0.0780393i	$0.0248660\pi$
$821$	4.47214i	0.156079i	−0.996950	+	0.0780393i	$0.975134\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	4.47214i	0.155794i
$825$	0	0
$826$	69.2820i	2.41063i
$827$	17.8885i	0.622046i	0.950402	+	0.311023i	$0.100672\pi$
$827$	17.8885i	0.622046i	−0.950402	+	0.311023i	$0.899328\pi$
$828$	0	0
$829$	−13.0000	−0.451509	−0.225754	−	0.974184i	$-0.572485\pi$
$829$	−13.0000	−0.451509	−0.225754	+	0.974184i	$0.572485\pi$
$830$	− 22.3607i	− 0.776151i
$831$	0	0
$832$	0	0
$833$	19.3649	0.670955
$834$	0	0
$835$	−20.0000	−0.692129
$836$	−46.4758	−1.60740
$837$	0	0
$838$	− 34.6410i	− 1.19665i
$839$	17.8885i	0.617581i	0.951130	+	0.308791i	$0.0999242\pi$
$839$	17.8885i	0.617581i	−0.951130	+	0.308791i	$0.900076\pi$
$840$	0	0
$841$	−14.0000	−0.482759
$842$	−34.8569	−1.20125
$843$	0	0
$844$	−24.0000	−0.826114
$845$	0	0
$846$	0	0
$847$	31.1769i	1.07125i
$848$	11.6190	0.398996
$849$	0	0
$850$	0	0
$851$	13.4164i	0.459909i
$852$	0	0
$853$	5.19615i	0.177913i	0.996036	+	0.0889564i	$0.0283532\pi$
$853$	5.19615i	0.177913i	−0.996036	+	0.0889564i	$0.971647\pi$
$854$	54.2218	1.85543
$855$	0	0
$856$	− 17.3205i	− 0.592003i
$857$	34.8569	1.19069	0.595344	−	0.803471i	$-0.297016\pi$
$857$	34.8569	1.19069	0.595344	+	0.803471i	$0.297016\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	13.4164i	0.457496i
$861$	0	0
$862$	10.0000	0.340601
$863$	31.3050i	1.06563i	0.846231	+	0.532816i	$0.178866\pi$
$863$	31.3050i	1.06563i	−0.846231	+	0.532816i	$0.821134\pi$
$864$	0	0
$865$	34.6410i	1.17783i
$866$	− 78.2624i	− 2.65946i
$867$	0	0
$868$	0	0
$869$	35.7771i	1.21365i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	60.0000	2.02953
$875$	−38.7298	−1.30931
$876$	0	0
$877$	− 29.4449i	− 0.994282i	−0.867670	−	0.497141i	$-0.834383\pi$
$877$	− 29.4449i	− 0.994282i	0.867670	−	0.497141i	$-0.165617\pi$
$878$	− 4.47214i	− 0.150927i
$879$	0	0
$880$	−10.0000	−0.337100
$881$	27.1109	0.913389	0.456694	−	0.889624i	$-0.349033\pi$
$881$	27.1109	0.913389	0.456694	+	0.889624i	$0.349033\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.4919	0.520168	0.260084	−	0.965586i	$-0.416250\pi$
$887$	15.4919	0.520168	0.260084	+	0.965586i	$0.416250\pi$
$888$	0	0
$889$	13.8564i	0.464729i
$890$	− 22.3607i	− 0.749532i
$891$	0	0
$892$	83.1384i	2.78368i
$893$	15.4919	0.518418
$894$	0	0
$895$	17.3205i	0.578961i
$896$	−54.2218	−1.81142
$897$	0	0
$898$	10.0000	0.333704
$899$	0	0
$900$	0	0
$901$	45.0000	1.49917
$902$	− 22.3607i	− 0.744529i
$903$	0	0
$904$	43.3013i	1.44018i
$905$	42.4853i	1.41226i
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	67.0820i	2.22620i
$909$	0	0
$910$	0	0
$911$	−46.4758	−1.53981	−0.769906	−	0.638157i	$-0.779697\pi$
$911$	−46.4758	−1.53981	−0.769906	+	0.638157i	$0.779697\pi$
$912$	0	0
$913$	−20.0000	−0.661903
$914$	−27.1109	−0.896748
$915$	0	0
$916$	62.3538i	2.06023i
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	−38.7298	−1.27688
$921$	0	0
$922$	−5.00000	−0.164666
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−23.2379	−0.763645
$927$	0	0
$928$	25.9808i	0.852860i
$929$	− 55.9017i	− 1.83408i	−0.398801	−	0.917038i	$-0.630573\pi$
$929$	− 55.9017i	− 1.83408i	0.398801	−	0.917038i	$-0.369427\pi$
$930$	0	0
$931$	17.3205i	0.567657i
$932$	0	0
$933$	0	0
$934$	− 51.9615i	− 1.70023i
$935$	−38.7298	−1.26660
$936$	0	0
$937$	−1.00000	−0.0326686	−0.0163343	−	0.999867i	$-0.505200\pi$
$937$	−1.00000	−0.0326686	−0.0163343	+	0.999867i	$0.505200\pi$
$938$	− 26.8328i	− 0.876122i
$939$	0	0
$940$	−30.0000	−0.978492
$941$	4.47214i	0.145787i	0.997340	+	0.0728937i	$0.0232234\pi$
$941$	4.47214i	0.145787i	−0.997340	+	0.0728937i	$0.976777\pi$
$942$	0	0
$943$	17.3205i	0.564033i
$944$	8.94427i	0.291111i
$945$	0	0
$946$	20.0000	0.650256
$947$	17.8885i	0.581300i	0.956830	+	0.290650i	$0.0938715\pi$
$947$	17.8885i	0.581300i	−0.956830	+	0.290650i	$0.906129\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	−30.0000	−0.972306
$953$	−15.4919	−0.501833	−0.250916	−	0.968009i	$-0.580732\pi$
$953$	−15.4919	−0.501833	−0.250916	+	0.968009i	$0.580732\pi$
$954$	0	0
$955$	34.6410i	1.12096i
$956$	− 13.4164i	− 0.433918i
$957$	0	0
$958$	40.0000	1.29234
$959$	−54.2218	−1.75091
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	57.1577i	1.84092i
$965$	−3.87298	−0.124676
$966$	0	0
$967$	− 10.3923i	− 0.334194i	−0.985940	−	0.167097i	$-0.946561\pi$
$967$	− 10.3923i	− 0.334194i	0.985940	−	0.167097i	$-0.0534393\pi$
$968$	− 20.1246i	− 0.646830i
$969$	0	0
$970$	− 34.6410i	− 1.11226i
$971$	30.9839	0.994320	0.497160	−	0.867659i	$-0.334376\pi$
$971$	30.9839	0.994320	0.497160	+	0.867659i	$0.334376\pi$
$972$	0	0
$973$	− 27.7128i	− 0.888432i
$974$	85.2056	2.73016
$975$	0	0
$976$	7.00000	0.224065
$977$	11.1803i	0.357691i	0.983877	+	0.178845i	$0.0572362\pi$
$977$	11.1803i	0.357691i	−0.983877	+	0.178845i	$0.942764\pi$
$978$	0	0
$979$	−20.0000	−0.639203
$980$	− 33.5410i	− 1.07143i
$981$	0	0
$982$	− 86.6025i	− 2.76360i
$983$	− 8.94427i	− 0.285278i	−0.989775	−	0.142639i	$-0.954441\pi$
$983$	− 8.94427i	− 0.285278i	0.989775	−	0.142639i	$-0.0455588\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	33.5410i	1.06816i
$987$	0	0
$988$	0	0
$989$	−15.4919	−0.492615
$990$	0	0
$991$	14.0000	0.444725	0.222362	−	0.974964i	$-0.428623\pi$
$991$	14.0000	0.444725	0.222362	+	0.974964i	$0.428623\pi$
$992$	0	0
$993$	0	0
$994$	− 34.6410i	− 1.09875i
$995$	49.1935i	1.55954i
$996$	0	0
$997$	29.0000	0.918439	0.459220	−	0.888323i	$-0.348129\pi$
$997$	29.0000	0.918439	0.459220	+	0.888323i	$0.348129\pi$
$998$	−46.4758	−1.47117
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.g.1351.1		4
3.2	odd	2	inner	1521.2.b.g.1351.3		4
13.3	even	3		117.2.q.d.82.2	yes	4
13.4	even	6		117.2.q.d.10.2	yes	4
13.5	odd	4		1521.2.a.u.1.2		4
13.8	odd	4		1521.2.a.u.1.3		4
13.12	even	2	inner	1521.2.b.g.1351.4		4
39.5	even	4		1521.2.a.u.1.4		4
39.8	even	4		1521.2.a.u.1.1		4
39.17	odd	6		117.2.q.d.10.1	✓	4
39.29	odd	6		117.2.q.d.82.1	yes	4
39.38	odd	2	inner	1521.2.b.g.1351.2		4
52.3	odd	6		1872.2.by.l.433.1		4
52.43	odd	6		1872.2.by.l.1297.2		4
156.95	even	6		1872.2.by.l.1297.1		4
156.107	even	6		1872.2.by.l.433.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.q.d.10.1	✓	4	39.17	odd	6
117.2.q.d.10.2	yes	4	13.4	even	6
117.2.q.d.82.1	yes	4	39.29	odd	6
117.2.q.d.82.2	yes	4	13.3	even	3
1521.2.a.u.1.1		4	39.8	even	4
1521.2.a.u.1.2		4	13.5	odd	4
1521.2.a.u.1.3		4	13.8	odd	4
1521.2.a.u.1.4		4	39.5	even	4
1521.2.b.g.1351.1		4	1.1	even	1	trivial
1521.2.b.g.1351.2		4	39.38	odd	2	inner
1521.2.b.g.1351.3		4	3.2	odd	2	inner
1521.2.b.g.1351.4		4	13.12	even	2	inner
1872.2.by.l.433.1		4	52.3	odd	6
1872.2.by.l.433.2		4	156.107	even	6
1872.2.by.l.1297.1		4	156.95	even	6
1872.2.by.l.1297.2		4	52.43	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{2} -3.00000 q^{4} -2.23607i q^{5} -3.46410i q^{7} +2.23607i q^{8} -5.00000 q^{10} +4.47214i q^{11} -7.74597 q^{14} -1.00000 q^{16} -3.87298 q^{17} -3.46410i q^{19} +6.70820i q^{20} +10.0000 q^{22} -7.74597 q^{23} +10.3923i q^{28} +3.87298 q^{29} +6.70820i q^{32} +8.66025i q^{34} -7.74597 q^{35} -1.73205i q^{37} -7.74597 q^{38} +5.00000 q^{40} -2.23607i q^{41} +2.00000 q^{43} -13.4164i q^{44} +17.3205i q^{46} +4.47214i q^{47} -5.00000 q^{49} -11.6190 q^{53} +10.0000 q^{55} +7.74597 q^{56} -8.66025i q^{58} -8.94427i q^{59} -7.00000 q^{61} +13.0000 q^{64} +3.46410i q^{67} +11.6190 q^{68} +17.3205i q^{70} +4.47214i q^{71} -15.5885i q^{73} -3.87298 q^{74} +10.3923i q^{76} +15.4919 q^{77} +8.00000 q^{79} +2.23607i q^{80} -5.00000 q^{82} +4.47214i q^{83} +8.66025i q^{85} -4.47214i q^{86} -10.0000 q^{88} +4.47214i q^{89} +23.2379 q^{92} +10.0000 q^{94} -7.74597 q^{95} +6.92820i q^{97} +11.1803i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{4} - 20 q^{10} - 4 q^{16} + 40 q^{22} + 20 q^{40} + 8 q^{43} - 20 q^{49} + 40 q^{55} - 28 q^{61} + 52 q^{64} + 32 q^{79} - 20 q^{82} - 40 q^{88} + 40 q^{94}+O(q^{100}) \) 4 * q - 12 * q^4 - 20 * q^10 - 4 * q^16 + 40 * q^22 + 20 * q^40 + 8 * q^43 - 20 * q^49 + 40 * q^55 - 28 * q^61 + 52 * q^64 + 32 * q^79 - 20 * q^82 - 40 * q^88 + 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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